Properties

Label 87.4.i.a
Level $87$
Weight $4$
Character orbit 87.i
Analytic conductor $5.133$
Analytic rank $0$
Dimension $96$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [87,4,Mod(4,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.4");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 87.i (of order \(14\), degree \(6\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.13316617050\)
Analytic rank: \(0\)
Dimension: \(96\)
Relative dimension: \(16\) over \(\Q(\zeta_{14})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 96 q + 52 q^{4} + 28 q^{5} + 12 q^{6} - 40 q^{7} + 144 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 96 q + 52 q^{4} + 28 q^{5} + 12 q^{6} - 40 q^{7} + 144 q^{9} - 140 q^{11} + 304 q^{13} - 294 q^{15} - 692 q^{16} + 800 q^{20} - 528 q^{22} + 324 q^{23} - 54 q^{24} - 170 q^{25} - 1288 q^{26} - 956 q^{28} - 478 q^{29} + 1056 q^{30} - 616 q^{31} - 406 q^{32} + 72 q^{33} + 34 q^{34} + 1032 q^{35} - 468 q^{36} + 1792 q^{37} + 1912 q^{38} + 2688 q^{40} + 762 q^{42} - 1092 q^{43} + 1750 q^{44} + 630 q^{45} + 2772 q^{47} + 1848 q^{48} - 1304 q^{49} - 882 q^{50} - 192 q^{51} + 1304 q^{52} - 3066 q^{53} - 108 q^{54} - 1974 q^{55} - 4704 q^{56} - 1752 q^{57} - 6092 q^{58} - 640 q^{59} - 7812 q^{60} - 252 q^{61} - 532 q^{62} - 18 q^{63} - 454 q^{64} - 8346 q^{65} + 5628 q^{66} + 744 q^{67} + 5530 q^{68} + 2100 q^{69} + 3216 q^{71} + 2142 q^{72} + 6510 q^{73} + 5090 q^{74} + 6594 q^{76} - 8876 q^{77} - 2118 q^{78} + 7092 q^{80} - 1296 q^{81} - 10590 q^{82} + 572 q^{83} - 2520 q^{85} + 12680 q^{86} - 2238 q^{87} + 6472 q^{88} - 2800 q^{89} + 1576 q^{91} - 10418 q^{92} + 60 q^{93} - 17564 q^{94} + 1374 q^{96} + 11690 q^{97} + 22554 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4.1 −5.32561 1.21553i −1.30165 2.70291i 19.6768 + 9.47585i 1.36680 5.98834i 3.64660 + 15.9768i −25.4953 + 12.2779i −59.1063 47.1357i −5.61141 + 7.03648i −14.5581 + 30.2302i
4.2 −5.09551 1.16302i 1.30165 + 2.70291i 17.4039 + 8.38126i −2.72939 + 11.9583i −3.48905 15.2865i 11.6156 5.59376i −46.2437 36.8782i −5.61141 + 7.03648i 27.8153 57.7591i
4.3 −3.92262 0.895312i −1.30165 2.70291i 7.37761 + 3.55287i −1.96973 + 8.62997i 2.68594 + 11.7679i 21.0311 10.1281i −0.593065 0.472954i −5.61141 + 7.03648i 15.4530 32.0886i
4.4 −3.58771 0.818871i 1.30165 + 2.70291i 4.99335 + 2.40467i 1.45209 6.36202i −2.45661 10.7631i −16.7086 + 8.04642i 7.07137 + 5.63923i −5.61141 + 7.03648i −10.4193 + 21.6360i
4.5 −2.51168 0.573274i −1.30165 2.70291i −1.22787 0.591312i 4.17965 18.3122i 1.71982 + 7.53503i −3.13542 + 1.50994i 18.8587 + 15.0393i −5.61141 + 7.03648i −20.9958 + 43.5983i
4.6 −2.42711 0.553971i 1.30165 + 2.70291i −1.62379 0.781976i 0.563333 2.46813i −1.66191 7.28132i 6.69997 3.22654i 19.0790 + 15.2150i −5.61141 + 7.03648i −2.73454 + 5.67833i
4.7 −1.72017 0.392617i −1.30165 2.70291i −4.40292 2.12033i −1.38401 + 6.06376i 1.17785 + 5.16050i −7.07166 + 3.40553i 17.7770 + 14.1767i −5.61141 + 7.03648i 4.76148 9.88731i
4.8 0.0265736 + 0.00606526i 1.30165 + 2.70291i −7.20708 3.47075i −0.0451194 + 0.197681i 0.0181958 + 0.0797209i −15.3287 + 7.38191i −0.340951 0.271899i −5.61141 + 7.03648i −0.00239797 + 0.00497944i
4.9 1.30198 + 0.297169i −1.30165 2.70291i −5.60090 2.69725i −4.08818 + 17.9115i −0.891506 3.90594i 1.77437 0.854493i −14.8436 11.8374i −5.61141 + 7.03648i −10.6455 + 22.1056i
4.10 1.49974 + 0.342306i 1.30165 + 2.70291i −5.07570 2.44433i 4.74328 20.7817i 1.02692 + 4.49922i 10.9198 5.25869i −16.3971 13.0763i −5.61141 + 7.03648i 14.2274 29.5434i
4.11 1.76854 + 0.403657i −1.30165 2.70291i −4.24297 2.04330i 1.80833 7.92279i −1.21097 5.30561i −26.4428 + 12.7342i −18.0251 14.3745i −5.61141 + 7.03648i 6.39618 13.2818i
4.12 2.00965 + 0.458689i 1.30165 + 2.70291i −3.37946 1.62746i −3.83335 + 16.7950i 1.37607 + 6.02895i −24.5544 + 11.8248i −18.9379 15.1025i −5.61141 + 7.03648i −15.4074 + 31.9937i
4.13 2.73476 + 0.624190i −1.30165 2.70291i −0.118474 0.0570538i 0.858190 3.75998i −1.87257 8.20427i 27.5583 13.2714i −17.8332 14.2215i −5.61141 + 7.03648i 4.69388 9.74694i
4.14 3.86408 + 0.881950i 1.30165 + 2.70291i 6.94550 + 3.34478i −2.23984 + 9.81340i 2.64585 + 11.5922i 26.9760 12.9910i −0.901956 0.719286i −5.61141 + 7.03648i −17.3099 + 35.9443i
4.15 4.81413 + 1.09879i 1.30165 + 2.70291i 14.7607 + 7.10840i 1.38037 6.04778i 3.29638 + 14.4424i −11.0304 + 5.31195i 32.3644 + 25.8098i −5.61141 + 7.03648i 13.2905 27.5981i
4.16 4.87893 + 1.11358i −1.30165 2.70291i 15.3561 + 7.39513i 3.82231 16.7467i −3.34075 14.6368i 0.370677 0.178508i 35.3857 + 28.2191i −5.61141 + 7.03648i 37.2976 77.4493i
13.1 −2.41931 + 5.02376i −2.34549 1.87047i −14.3971 18.0534i −5.35025 2.57654i 15.0713 7.25794i −10.9950 + 13.7872i 82.0381 18.7247i 2.00269 + 8.77435i 25.8879 20.6449i
13.2 −2.22145 + 4.61288i 2.34549 + 1.87047i −11.3559 14.2399i −10.0741 4.85142i −13.8386 + 6.66434i 21.3564 26.7801i 50.9810 11.6361i 2.00269 + 8.77435i 44.7580 35.6933i
13.3 −1.93516 + 4.01839i 2.34549 + 1.87047i −7.41472 9.29777i 11.6028 + 5.58764i −12.0552 + 5.80547i −13.8901 + 17.4176i 16.9247 3.86294i 2.00269 + 8.77435i −44.9066 + 35.8118i
13.4 −1.78908 + 3.71506i −2.34549 1.87047i −5.61296 7.03842i 16.3624 + 7.87972i 11.1452 5.36724i 21.8101 27.3490i 4.03000 0.919821i 2.00269 + 8.77435i −58.5473 + 46.6899i
See all 96 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 87.4.i.a 96
29.e even 14 1 inner 87.4.i.a 96
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
87.4.i.a 96 1.a even 1 1 trivial
87.4.i.a 96 29.e even 14 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(87, [\chi])\).